Construction of axial-flow turbine blades

ABSTRACT

Axial-flow turbine nozzles and moving blades which employ hollow blade means, the blades having a wall thickness distribution on both sides, suction side and pressure side, of the hollow portion which is selected in accordance with the distribution of effective local heat transfer coefficients along the blade surface in the chordwise direction, whereby the temperature distribution in said hollow blades responds almost uniformly to the temperature change of the motive fluid.

llnited States Patent [191 Matstilti et al.

[ (IUNSTRUCTION 0F AXIAL-FLOW TURBINE BLADES [75] Inventors: MasakatsuMatsuki; Toyoaki Yoshida, both of Tokyo, Japan [73] Assignee: TheDirector of National Aerospace Laboratory of Science and TechnologyAgency, Masao Yamanouchi, Tokyo, Japan [22] Filed: Dec. 11, 1972 [21]Appl. No.: 313,843

[30] Foreign Application Priority Data May 8, 1972 Japan 47-44663 [52]US. Cl. 416/96, 415/115 [51] Int. Cl. F01d 5/18 [58] Field of Search416/92, 95-97; 415/1 15-] 16 [56] References Cited UNITED STATES PATENTS3,420,502 1/1969 Howald 416/96 X 11 3,836,283 1 Sept. 17, 1974 3,650,6353/1972 Wachtell et a1 416/97 UX FOREIGN PATENTS OR APPLICATIONS 924,0123/1947 France 416/96 892,698 10/1953 Germany 416/96 910,400 1 1/1962Great Britain 416/92 Primary Examiner-Everette A. Powell, Jr. Attorney,Agent, or FirmBrooks Haidt & Haffner ABSTRACT 4 Claims, 8 DrawingFigures SHEET 1 [If a FIG,

TRAILING SIDE EDGE OZCX LEADING SUCTION SIDE EDGE TRAILING PRESSURE EDGEO O m O O O 2 mnzmwsfimlw 3.836.283 E SHEET 2 {If H O y l I l I I I I l1 (sec) (mm) o TRAILING PRESSURE LEADING SUCTION TRAILING EDGE SIDE EDGESIDE EDGE THICKNESS CONSTRUCTION OF AXIAL-FLOW TURBINE BLADES BACKGROUNDOF THE INVENTION l. Field of the Invention Y The present inventionrelates to improvements in turbine blades and nozzles and moreparticularly to a construction of axial-flow turbine blades for a gasturbine or a steam turbine which is subjected to frequent starts andstops.

2. Description of the Prior Art Much research on fluid cooled turbineblades has been carried out and many inventions have been made. However,almost all of such research and inventions have been intended to makethe blade temperature uniform and to keep it lower under steady stateconditions. Effective cooling methods of a turbine blade which issubject to severe thermal conditions which have been adoptedsuccessfully are impingement cooling or film cooling at the leading edgeand film cooling at the trailing edge. Therefore, heat resistance of theblade has been considerably improved under steady state conditions.Generally speaking, heat capacities at the leading edge and the trailingedge are relatively small compared with heat capacities in the chordwisedirection of the middle part of the blade. Therefore, if the blade issubject to a sudden change of the temperature of the motive fluid, eachpart of the blade shows different response and excessive thermalstresses come about at the leading edge and/or the trailing edge, andfor such reason, many examples of blades with cracks are found.

SUMMARY OF THE INVENTION It is an object of the present invention toeliminate the above-mentioned disadvantage existing in the conventionalform of turbine blade and to provide a hollow turbine blade, the wallthickness distribution of which corresponds to the effective local heattransfer coefficient distribution of said blade. Similar principles areapplicable to turbine nozzles.

It is another object of the present invention to provide a method fordetermining the turbine blade thickness distribution.

It is a further object of the present invention to provide a method forreducing an effective local heat transfer coefficient, in the case thatthe blade thickness is restricted by the aerodynamic performances of theblade at the trailing edge region.

These and other objects of the present invention will be apparent whenthe reference is made to the following description and accompanyingdrawings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. l is a cross-sectional, end viewof an axial-flow turbine blade constructed in accordance with thepresent invention, and the cooling thereof comprises impingement coolingat the leading edge, convection cooling in the mid-chord region and filmcooling at the trailing edge;

FIG. 2 is a graph illustrating the distribution of the effective localheat transfer coefficients;

FIG. 3 is a schematic diagram used for the calculation of the non-steadystate, one-dimensional temperature;

FIG. 4 is a graph illustrating the non-steady state temperaturedistribution at the leading edge of a blade with elapsed time;

FIG. 5 is a graph of the blade thickness distribution in the chordwisedirection, i.e., leading edge to trailing edge direction, of the outershell of the blade;

FIG. 6 is a cross-sectional, end view of another example of anaxial-flow turbine blade constructed in accor dance with the presentinvention, in which film cooling is used at the leading edge region andthe trailing edge region;

FIG. 7 is a graph illustrating the distribution, of nonsteady statethermal stresses without the use of the invention; and

FIG. 8 is a graph illustrating the distribution of nonsteady statethermal stresses with the use of the present invention.

DETAILED DESCRIPTION OF THE INVENTION The present invention provides amethod for making the shell thickness distribution on the pressure sideand the suction side of the hollow blade correspond to the distributionof the effective local heat transfer coefficients along the bladesurface in the chordwise direction, i.e., the direction from the leadingedge to the trailing edge. According to the method, the temperature ateach part of the blade changes almost uniformly even in the case of thetransient operation such as starting, stopping, acceleration anddeceleration. Therefore, no excessive thermal stresses occur in theblade, and consequently, the durability of the blade constructed by suchmethod is remarkably increased compared with that of the conventionalhollow blade which is constructed without taking into consideration thetransient operation.

The said effective local heat transfer coefficient a is a constant ofproportionality, defined by the following equation,

where q represents heat flux, Tg represents the recovery temperature ofa motive fluid and Tb represents the blade surface temperature. In thecase of film cooling and transpiration cooling, the local heat transfercoefficient a is expressed as follows,

q 0:(Taw Tb) where Taw represents adiabatic wall temperature of theblade. It will be noted that with the cooling of the turbine blade bysecondary fluid, Tg is always higher than Taw. Therefore, from equations(1) and (2) one obtains the relation that a a. This relation means thatif (Tg Tb) is introduced as a standard temperature difference even inthe case of a film cooling or a transpiration cooling, the value ofeffective local heat transfer coefficient is lower than the local heattransfer coefficient, whereas the effective local heat transfercoefficient a coincides with the conventional local heat transfercoefficient without secondary air cooling. Therefore, the factors can betaken into account by equation (1) independently of the cooling methods.

In what follows, theoretical background, blade construction and effectsof the present invention are given together with the description of thefigures.

The blade illustrated in cross-section in FIG. 1 has an outer shell 1having a lower pressure or suction surface wall and a high pressure orpressure surface wall, the suction side being designated by the numeral3 and the pressure side being designated by the numeral 4, and a coolingfluid insert or duct 2 is' within the shell 1 and has its outer wallspaced from the inner wall of the shell 1. The insert 2 has an opening2a for directing cooling fluid against the leading edge portion of theshell 1, and the fluid flows rearwardly of the blade between the outerwall of the insert 2 and the inner wall of the shell 1 and is exhaustedthrough the channel 1a.

Reference numeral 5 designates one of the small elements or portions ofthe outer shell 1 which is used for the application of numericalcalculations. Reference numerals 6 and 7 designate main air flow sideand cooling air flow side of the hollow blade respectively.

In FIG. 1, the intersections of the extensions of the wall ofthe'impingement hole 2a of said insert 2 and the inner surface of thesaid outer shell 1 is designated by the letter P. Extensions which are 650 on both sides of the impingement hole center line and which gothrough the center of the circle which contains the blade leading edgewill intersect the inner surface of the said outer shell 1 at Q. Themain flow is divided into two parts, suction side surface flow andpressure side surface flow, at the outer surface stagnation point R. Onthe other hand, the cooling air impinges on the inner surface stagnationpoint S which is located at the inner side of the shell 1, and oppositeto the point R.

If there occurs a sudden temperature change of the motive fluidimpinging on convection cooled turbine blades, such as in FIG. 1, almostall of the heat flow is transferred in the shell thickness direction.Therefore, the heat flow by conduction, both in the chordwise directionand in the spanwise direction, can be neglected. Consequently,non-steady state temperature in said small element 5 is obtainedanalytically from the fundamental equation for one-dimensional,non-steady state heat conduction, for which the distribution of thelocal heat transfer coefficients and the temperature distribution in theambient fluid are needed as boundary conditions. Incidentally, theheat-flow by conduction both in the chordwise direction and in thespanwise direction are neglected in the present calculations, but ifthese heat flows are also considered in-determining the temperaturedistribution, an even more effective blade will be realized.

The local heat transfer coefficients a in the main air flow side 6 alongthe outer surface of the shell 1 and a in the cooling air flow side 7along the inner surface of the shell 1, in the chordwise direction canbe calculated from the empirical equations explained below. Theempirical equation on the convective heat transfer is univers'allydescribed with some dimensionless numbers as follows,

Nut R m P a proper textbook of Heat Transfer, e.g., Heat & Mass Transferby Eckert, Drake, McGraw-Hill, or Heat Transmission by McAdams',McGraw-Hill, and c, m and n are numerical constants. Then, the localheat transfer coefficient a, can be obtained by substituting Nu,a,-X/)t, and Re, =-UX/v into equation (3) and adopting the values of c,m and n which are suitable to the portion of the blade surfaceconsidered, where at represents the heat transfer coefficient, X standsfor a representative length, )t represents the thermal conductivity ofthe fluid, U represents the velocity of fluid and 1/ represents thekinetic viscosity of fluid.

According to this procedure, the values of the heat transfercoefficients are calculated from each empirical equation applied to theblade portion identified hereinafter.

a. Main air flow side (a i. the leading edge stagnation point R and itsneiborhood region:

In the case of the turbine blade under consideration, the leading edgeregion can be considered as a circular cylinder in the'range from theleading edge stagnation point R to 0 Therefore, the following empiricalequation by Schmidt and Wenner (see Forschung, 12, (1941)) is used forthe heat transfer coefficients along the circumference of a circularcylinder,

(4) with 0 6 s 60, where )t represents a thermal con- Kgusroa X0.5

with

y a60 s60 [U ll where U represents the local velocity of the main airflow, X represents the distance from the leading edge stagnation pointalong the blade surface in the chordwise direction, a represents a heattransfer coefficient at the point 6 60, and U represents the main airflow velocity at the point 0 60. When the transition point is reached onthe blade surface, the following empirical equation is adopted in therearward direction from the transition point;

This equation is derived from the equation for the turbulent boundarylayer along a flat plate. b. Cooling air flow side (01) i. the leadingedge stagnation point S and the adjacent region: The heat transfercoefficient a at the leading edge stagnation point S in the cooling airflow side is obwhere S and 5, represent surface heat transfer area inthe main air flow side and in the cooling air flow side, respectively. Trepresents the main air flow inlet temperature and T represents thecooling air inlet temperature. is obtained from equation (4). The valueof a is applied to the region from the stagnation point S to the point Pshown in FIG. 1.

ii. the region adjacent to the leading edge area:

The stream flow of the cooling air in the region adjacent to the leadingedge area can be considered to be equivalent to that of the jet flowwhich impinges upon a flat plate. Therefore, the following empiricalequation is applied to the region from the point P to the point Q shownin FIG. ll:

where U represents the local velocity of the cooling air flow, 1:,-represents the distance in the chordwise direction from the leading edgestagnation point S in the cooling air flow side along the inner surfaceof the outer shell ll, and U and X are values of U and X at the point P,respectively.

iii. the mid-chord region and trailing edge region:

In the region of the blade which is rearward from the point Q, theempirical equation for the turbulent boundary layer along a flat plateis applied,

where a U and X are values of a U and X at the point Q, respectively.Making use of equations (4) (9), the local heat transfer coefficients a,and a were calculated for the turbine blade shown in FIG. I under thefollowing conditions: turbine inlet temperature T l,l50 C, cooling airinlet temperature T 500 C, main flow inlet velocity U l 14 m/sec andcooling air weight flow ratio Wc/Wg= 2 percent, where We is cooling airweight flow rate and Wg is main air weight flow rate.

FIG. 2 is a graph which shows the distribution of the blade surfacelocal heat transfer coefficients a a using the methods of calculationjust described. In this figure the ordinate is the heat transfercoefficient, the abscissa is the distance along the outer blade surfaceand the origin corresponds to the leading edge stagnation points R andS.

FIG. 3 is a schematic diagram referred to for the calculation of thenon-steady state temperature in a small element of the blade, such asthe small element 5. Let

the blade shell thickness be I, and assume that the y axis is orientedin the blade shell thickness direction with its origin located at theblade surface in the main air flow side. T, represents the main air flowtemperature, and T, represents cooling air flow temperature. T, and Trepresents the blade surface temperatures at y 0 and y 1 respectively,under steady state conditions. To represents the temperature of theentire region kept in an equilibrium state that is realized beforeheating or after cooling. Temperature T(y,t) at an arbitrary positionand arbitrary time in the small element can be obtained from thefollowing fundamental equation:

where t is the elapsed time after a sudden temperature change of themotive fluid and a represents the thermal diffusivity of the bladematerial. The analytical solution of equation (10) with the followingboundary conditions and initial conditions is already set forth in anarticle by l. Fujii and N. Isshiki appearing in Vol. 35 No. 271 forMarch 1969 of the publication TRANSAC- TIONS OF THE J.S.M.E.

Boundary conditions and initial conditions: In the case of heating (H)at y O,

a, (T T(0,t)) ()t6T/8y) y 0 aty=l,

a,.,(T(l,r) T,.) ()t8T/8y) y =1 at I O,

T(y,O) =T0 at t (y A A TB)y/l In the case of cooling (C) at y O,

1( o) y) y 0 at y =1,

WATUJ) o) PAST/ y) y at I O,

(y TA (TA TB )y/l at z where A represents the thermal conductivity ofthe blade material, and T and T are described as follows,

The results for the non-steady state, one-dimensional temperaturedistributions are then: In the case of heating (H) where T representsthe non-steady term of the temperature, and is expressed as follows,

w T 2 2 (F n Il= 1 a cos (any) K,, sin (a y) (a K )l+ (Kc+ K (01,, KCK(ozn K6 where C C K and K, are constants defined by the followingequations,

1 TA 2 (TA T g a /k and K a /A and a is a positive root in the equation:

First of all, the non-steady state temperature at the leading edge (1:O,y O) was calculated under the following conditions: T l, 1 50 C, T,.500 C, a 4.44 X 10 m /sec., Kcallmh C, a 1,360 Kcallm h C, a 1,520Kcal/m h C, l 2mm and chord length 32 mm. Then, the calculations werecarried out also at the point (x O,y N2) and (x O,y =1) according to thesame procedure. These results are plotted in FIG. 4 where the ordinateis dimensionless temperature T T /T T,, the abscissa is elapsed time t,and the symbols (H) and (C) correspond to the case of heating andcooling respectively. As is evident from equations 15) l7) and FIG. 4,the response of temperature T(y,t) is not exactly the same as thefirst-order response to the step input used in the linear dynamicsystem, but its trend is very similar. The time constant 1' of the bladetemperature T(y,t) is defined by the same method as is used in saidfirst-order response, namely is the elapsed time when 63.2 percent ofthe value at the steady state, T(y,), is reached.

If the transient temperature response were the same in every part of theblade, thermal stresses which come about under the transient operatingconditions can be considerably reduced. In order to reduce the thermalstresses, it is very effective to make the shell thickness 1 along theblade surface so that the time constant 7 may be considered much thesame in every part of the blade. Let the arithmetic mean value of timeconstants calculated at y =0, y =l/2 and y =l of the leading edge be arepresentative time constant rm. Then, the blade thickness at eachposition in the chordwise direction is determined so that its timeconstant will be equal to 1m, where the time constant at each positionis also the arithmetic mean value of the three points y O, y [/2 and yl.

The blade thickness distribution in the chordwise direction calculatedby the said procedure is shown by the graph of FIG. 5. In this figure,the ordinate is the blade thickness 1, the abscissa is the distancealong the blade surface and the origin corresponds to the leading edge.In this case, the representative time constant rm is equal to 2.391 sec.

FIG. 7 and FIG. 8 are graphs illustrating the distribution of non-steadystate thermal stresses obtained from non-steady state blade temperaturedistribution calculated by equations (13) (17). In these figures, the ordinate is the thermal stress aKg/mm the abscissa is the distance alongthe blade surface, and the elapsed time t is taken as the parameter.FIG. 7 is the result obtained in the case of constant blade thicknessthat does not take into consideration the desirability of equal timeconstants. On the other hand, FIG. 8 is a graph of the results obtainedby the methods of the present invention which considers the timeconstants and makes them substantially equal. From these two figures, itis apparent that if the transient response at every part of the blade istaken into consideration, thermal stresses can be remarkably reduced.Then, according to the present invention, crack initiation on the bladesurface can be avoided for far longer times than have heretofore beenaccomplished, and consequently, the blade can sufficiently withstand thefrequent starts and stops of the engines including the blades.

Further, in the event that the blade thickness calculated by the methodsof the present invention conflict with the blade profile designed on thebasis of the aerodynamic performance, especially in the trailing edgeregion, it is sufficient to make the effective local heat transfercoefficient correspond to the profile desired from the aerodynamicperformance and then introducing a film cooling or a transpirationcooling to the relevant region.

FIG. 6 is another embodiment of the turbine blades to which the presentinvention is applied. The cooling thereof comprises impingement coolingand film cooling at the leading edge, convection cooling in the midchordregion and film cooling in the trailing edge region. The outer shell 1encloses a pair of inserts 2b and 2c. The holes 8 and 9 are made at theleading edge region for film cooling. In this embodiment, the equalityof transient response of the various portions of the blade is easilyrealized within the required blade profile because of the application ofthe film cooling through the channels or holes 10 and 11 at the trailingedge.

In the embodiment shown in FIG. 1, heat flow by conduction both in thechordwise direction (x) and (-Adspanwise direction (2) is considerednegligibly small when the calculation of the non-steady statetemperature is carried out. However, if these heat flows are taken intoaccount, then the fundamental equation (10) should be modified in thefollowing manner.

stress does not occur in the blade. For this reason, axial-flow turbineblades constructed in accordance with the present invention are verystrong and resistant to frequent heat variations, such as by reason ofstarts and stops. In other words, the durability of the blade isremarkably increased.

Moreover, in accordance with the present invention, the turbine inlettemperature of the motive fluid can be higher, resulting in improvementof the thermal efficiency of a gas turbine or a steam turbine.

The construction of axial-flow turbine blades in accordance with thepresent invention is useful not only in aircraft engines, but also inmarine turbines, steam turbines, automobile engines, etc. Accordingly,the present invention is extremely useful for industrial purposes.

We claim:

l. A hollow turbine part for use in a hot fluid medium, said part havinga pressure surface wall and a suction surface wall and having a leadingedge and a trailing edge, said walls having a thickness distribution inthe direction from said leading edge to said trailing edge such that,with changes of the temperature of'said fluid, the temperature responseat each portion of said walls in substantially the same as thetemperature re sponse at the other portions of said walls, whereby thetemperature distribution in said walls changes substantially uniformlyin response to changes in temperature of said fluid.

2. A hollow turbine part as claimed in claim ll, wherein said part is ahollow blade and wherein said thickness distribution is such that eachportion of said walls has a mean temperature time constant which issubstantially equal to a predetermined time constant, said mean timeconstant at each portion of said walls being the mean of the temperaturetime constants at the outer surface thereof, at the inner surfacethereof and at an intermediate point between the surfaces thereof, eachof said outer surface, inner surface and intermediate point timeconstants being determined by replacing the temperature response at saidouter surface, said inner surface and said point to a step change ofsaid fluid temperature with approximately a first order responsethereto, and said predetermined time constant being substantially equalto the mean time constant at said leading edge of said blade.

3. A hollow turbine part as claimed in claim 1, further comprising meansfor supplying fluid cooling to at least one of said leading edge andsaid trailing edge to thereby lower the heat transfer coefficientthereof and modify the thickness thereof required to provide saidtemperature response therefor.

4. A hollow turbine blade for use in a fluid medium, said bladecomprising a pressure surface wall and a suction surface wall and havinga leading and trailing edge, said walls having a thickness distributionin the direction from the leading edge to the trailing edge of saidblade such that the temperature response at each portion of said wallsin substantially the same as the other portions of said walls withchanges of the temperature of said fluid and such that the meantemperature time constant is substantially equal to the mean temperaturetime constant at said leading edge of said blade, said mean timeconstant at each portion of said walls being the mean of the temperaturetime constants at the outer surface thereof, at the inner surfacethereof and at an intermediate point between the surfaces thereof, eachof said outer surface, inner surface and intermediate point timeconstants being determined by replacing the temperature response at saideach point to a step change of said fluid temperature with approximatelya first order response thereto, said temperature response at each pointbeing calculated from the following equa trons:

where T(y,t) represents a temperature at an arbitrary position and anarbitrary time in a small element of said wall, 1 represents elapsedtime after a sudden temperature change of said motive fluid, arepresents thermal diffusivity of the blade material, .and y representsthe axis oriented in the blade wall thickness direction with its originlocated at the blade surface in the main air flow side;

in the case of heating, the boundary conditions and initial conditionsare:

at y 0,

111( PAST/ y) y 0 at y l,

r-A M) 0) (-AfiT/Sy) y =l at t O,

T(y,0) T

at t= 05 2 TA (TA HY/ in the case of cooling, the boundary conditionsand initial conditions are:

at y 0,

a,,,(T(0,t) To) (MST/8y) y 0 at y =1,

a,. (T(I,t) T) (-AS'lf/By) y l at t= 0,

om A i B) y/l where in the boundary and initial condition equations Trepresents the recovery temperature of the fluid, T represents thecooling air flow temperature, To represents the temperature of theentire region kept in an equilibrium state that is realized beforeheating or after cooling, 1 represents the wall thickness and representsthe conductivity of the blade material, and T and T are:

TA 'i" H l/A (I y/ rr/ ux) T8 au cm c n mt 9: e a) m m'/ ux) where orrepresents an effective local heat transfer co efficient on main airflow side, or represents an effective local heat transfer coefficient onthe cooling air flow side; and where, in the case of heating:

03 TN TA (TA HU/I and in the case of cooling:

following equations:

W19 if inm t s where T represents the non-steady state term of the 1 oTA temperagtcure, and is expressed as follows: 5 C2 (TA TB) 2 2 K ne /A,and

a cos (a y) sin (a y) c m/ Q" V K "t and where oz is a positive root inthe equation X 1 2 2 n n Sin n l y n 2 C /a, +C K l/a,,cos (a,,l)+C K/a,,C /a,,} wn v a n g where C,, C K and K are constants defined by theTJNHTED STATES PATENT UFFTCE EERTTHQATE M @QRREQTTUN Patent 3,836,283Dated September 17, 1974 lnventofls) Masakatsu Matsuki and ToyoakiYoshida It is certified that error appears in the above-identifiedpatent and that said Letters Patent are hereby corrected as shown below:

In the drawing on the title page and in Fig. 1, add reference numeral---l-- to designate the blade outer shell.

Col. 3, line 5 "lower should read -low- Col. 4, Equation (4) shouldread:

0.,4 3 dg 1 l4 (lg/d1)Pr (U dl/vg 1 (6/90) Col 5, line 34 "X (2ndoccurrence) should read X Col. 6, Equation (10) should read:

--8'I/8t e a T a The equation in line 29 should read:

The equation in line 32 should read:

-d (T(-1,t)T ()\8T/8y) The equation in line 45 should read:

cont'd/ FORM PO-105O H0459] USCOMM-DC 60376-P69 v u s oovimmsm PnmnuoOFFICE I969 u-166334 UNITED STATES PATENT OFFICE CERHHQATE @l ECTIONPatent No. 3 r 836 r 283 Dat d eptember 17 1974 Page 2 Inventor(s';Masakatsu Matsuki and 'Tnvnsk-i V'nc'h-i as J rvu-L-LJ. LL

It is certified that error appears in the above-identified patent andthat said Letters Patent are hereby corrected as shown below:

Col. 6, The equation in line 48 should read:

-a (T(1,t T W line 65 insert closing parenthesis after (T /T (lstoccurrence) Col, 7, line 21 at the beginning of the line "fit" shouldread -x- The equation in line 30 should read: tan(oc (K K )OL /(OL K KThe terms in line 40 should read:

(T T vcr T Lines 60 & 62 "rm" should read --T Col. 8, line 4 "Tm" shouldread --T line 50 delete "(-l8" and insert in the- Equation (10) shouldread: BT/St a[(8 T/8x a w a (8 T/8z Col. 9, line 24 "in" should read -iscontd/ F ORM PO-105O (IO-69) USCOMM-DC 6O376-P69 U 5 GOVERNMENT PRINHNGDFFICE I969 0*]66-13 UNITED STATES PATENT oEEIcE CERTTHCATE 0FCORRECTTQN Patent No. 3 836,283 Dated September 17, 1974 Inventods')Masakatsu Matsuki and Toyoaki Yoshida Page 5 It is certified that errorappears in the above-identified patent and that said Letters Patent arehereby corrected as shown below:

Col. 10, the equation in line 8 should read:

-8T/8t a(8 'I'/8y The equation in line 21 should read:

The equation in line 24 should read:

1 (T( ,o T y) The equation in line 36 should read:

d (T (o,t) T

(l8T/By) The equation in line 39 should read:

x( I o) lines 55-59, the equations should read:

B l (T (OL L 1/[ cont'd/ USCOMM-DC 60376-5369 9 u s covimmznr PRINTINGOFFICE 19590-155-334 UNITED STATES PATENT OFFICE tmmtmt @l toEcnN PatentNo. 3 836 283 Dated September 17 1974 Inventor) Masakatsu Matsuki andToyoaki Page 4 It is certified that error appears in theabove-identified patent and that said Letters Patent are herebycorrected as shown below:

Col, 11, line 5 "-an t" should read aocit lines 11 & 12, the equationshould read:

-xC C 1 ,(C K /oci) 1 sin (ca t) (C K /oc (C /a (c Kgl/oc 1 cos (on l)(C Kg/oz (C /oc Col. 12, line 11 the equation should read:

2 -tan(d 1) (K +K on /(oc -K K and gtalcd tis thirtieth ay of September1975 [S Arrest" RUTH C. MASON CMARSHALL DANN AIHSIIHX Of'jireCommisxz'onw of Iuremx and Trademarks FORM PO-105O 110-69) USCQMM-DC60376-P69 n u s covzmmzm PRINTING OFHCE 19690-166434

1. A hollow turbine part for use in a hot fluid medium, said part havinga pressure surface wall and a suction surface wall and having a leadingedge and a trailing edge, said walls having a thickness distribution inthe direction from said leading edge to said trailing edge such that,with changes of the temperature of said fluid, the temperature responseat each portion of said walls in substantially the same as thetemperature response at the other portions of said walls, whereby thetemperature distribution in said walls changes substantially uniformlyin response to changes in temperature of said fluid.
 2. A hollow turbinepart as claimed in claim 1, wherein said part is a hollow blade andwherein said thickness distribution is such that each portion of saidwalls has a mean temperature time constant which is substantially equalto a predetermined time constant, said mean time constant at eachportion of said walls being the mean of the temperature time constantsat the outer surface thereof, at the inner surface thereof and at anintermediate point between the surfaces thereof, each of said outersurface, inner surface and intermediate point time constants beingdetermined by replacing the temperature response at said outer surface,said inner surface and said point to a step change of said fluidtemperature with approximately a first order response thereto, and saidpredetermined time constant being substantially equal to the mean timeconstant at said leading edge of said blade.
 3. A hollow turbine part asclaimed in claim 1, further comprising means for supplying fluid coolingto at least one of said leading edge and said trailing edge to therebylower the heat transfer coefficient thereof and modify the thicknessthereof required to provide said temperature response therefor.
 4. Ahollow turbine blade for use in a fluid medium, said blade comprising apressure surface wall and a suction surface wall and having a leadingand trailing edge, said walls having a thickness distribution in thedirection from the leading edge to the trailing edge of said blade suchthat the temperature response at each portion of said walls insubstantially the same as the other portions of said walls with changesof the temperature of said fluid and such that the mean temperature timeconstant is substantially equal to the mean temperature time constant atsaid leading edge of said blade, said mean time constant at each portionof said walls being the mean of the temperature time constants at theouter surface thereof, at the inner surface thereof and at anintermediate point between the surfaces thereof, each of said outersurface, inner surface and intermediate point time constants beingdetermined by replacing the temperature response at said each point to astep change of said fluid temperature with approximately a first orderresponse thereto, said temperature response at each point beingcalculated from the following equations: delta T/ delta t a( delta 2T/delta y2) where T(y,t) represents a temperature at an arbitrary positionand an arbitrary time in a small element of said wall, t representselapsed time after a sudden temperature change of said motive fluid, arepresents thermal diffusivity of the blade material, and y representsthe axis oriented in the blAde wall thickness direction with its originlocated at the blade surface in the main air flow side; in the case ofheating, the boundary conditions and initial conditions are: at y 0,Alpha gx(Tg - T(o,t)) (- lambda delta T/ delta y) y o at y l, Alphacx(T(l,t) - Tc) (- lambda delta T/y) y l at t 0, T(y,O) To at t Infinity, T(y, Infinity ) TA - (TA - TB) y/l in the case of cooling, theboundary conditions and initial conditions are: at y 0, Alphagx(T(o,t) - To) ( lambda delta T/ delta y) y o at y l, Alpha cx(T(l,t) -T) (- lambda delta T/ delta y) y l at t 0, T(y,0) TA - (TA - TB) y/l att Infinity , T(y, Infinity ) To where in the boundary and initialcondition equations Tg represents the recovery temperature of the fluid,Tc represents the cooling air flow temperature, To represents thetemperature of the entire region kept in an equilibrium state that isrealized before heating or after cooling, l represents the wallthickness and lambda represents the conductivity of the blade material,and TA and TB are: TA Tg(1 + Alpha cxl/ lambda + Alpha cx/ Alpha gxTc/Tg)/(1+ Alpha cxl/ lambda + Alpha cx/ Alpha gx) TB Tg(1 + Alpha cxl/lambda Tc/Tg + Alpha cx/ Alpha gx Tc/Tg)/(1+ Alpha cxl/ lambda + Alphacx/ Alpha gx) where Alpha gx represents an effective local heat transfercoefficient on main air flow side, Alpha cx represents an effectivelocal heat transfer coefficient on the cooling air flow side; and where,in the case of heating: T(y,t) TN + TA - (TA -TB)y/l and in the case ofcooling: T(y,t) To - TN where TN represents the non-steady state term ofthe temperature, and is expressed as follows: